Theory of vortex force microscopy in superconducting layers

Ona, Artorix de la Cruz de; Badía-Majós, A.

doi:10.1103/physrevb.70.144512

Cited by 4 publications

(6 citation statements)

References 30 publications

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“…MFM represents an optimal compromise among these approaches, providing a suitably high lateral spatial resolution (≥ 20 nm) to resolve a wide range of magnetic textures [41][42][43][44] while also enabling quantitative estimates of intrinsic magnetic properties of materials with judicious modeling of probe-sample geometry. [45][46][47][48] Although MFM is most often used to resolve magnetic fields originating from FM materials, it can also be used to resolve magnetic fields associated with induced, stray, or residual fields in AFM materials. [49][50][51][52] Despite the manifest utility of MFM for investigating magnetic behavior, its application to 2D materials has so far been minimal, being applied only to relatively thick layered magnetic materials that are effectively in their bulk magnetic state.…”

Section: Introductionmentioning

confidence: 99%

Visualizing Atomically Layered Magnetism in CrSBr

et al. 2022

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ability to synthesize and characterize 2D materials with long-range magnetic order was notably absent. Such materials hold great promise for potential applications in spintronics, [6][7][8][9][10][11] topological superconductivity, [12,13] vdW heterostructures, [14,15] and memory storage. [16,17] Hypothetical 2D magnets defy the Mermin-Wagner theorem, [18] which states that long-range magnetic order cannot exist at finite temperature within the 2D isotropic Heisenberg model. [19] Hence, magnetic anisotropy is a necessary prerequisite to realizing stable 2D magnetism, and has been demonstrated to exist in several layered materials over the last few years. [3,4,[20][21][22][23][24][25][26] Among these, the chromium trihalides CrX 3 (X = Cl, Br, I) have been the most extensively studied, all of which display intralayer ferromagnetic (FM) order and either FM or antiferromagnetic (AFM) interlayer coupling depending on the choice of halide and crystal thickness. [27] Despite their extensive characterization, chromium trihalides suffer from a lack of air and moisture stability and possess relatively low transition temperatures, limiting their applications. On the other hand, CrSBr has emerged [28][29][30][31] as an air-stable layered magnetic material possessing A-type AFM ordering with a comparatively high Néel temperature (T N ≈ 132 K).A variety of analytical tools have been used to characterize CrSBr, including magnetometry, [32,33] magneto-transport, [32,33] 2D materials can host long-range magnetic order in the presence of underlying magnetic anisotropy. The ability to realize the full potential of 2D magnets necessitates systematic investigation of the role of individual atomic layers and nanoscale inhomogeneity (i.e., strain) on the emergence of stable magnetic phases. Here, spatially dependent magnetism in few-layer CrSBr is revealed using magnetic force microscopy (MFM) and Monte Carlo-based simulations. Nanoscale visualization of the magnetic sheet susceptibility is extracted from MFM data and force-distance curves, revealing a characteristic onset of both intra-and interlayer magnetic correlations as a function of temperature and layer-thickness. These results demonstrate that the presence of a single uncompensated layer in odd-layer terraces significantly reduces the stability of the low-temperature antiferromagnetic (AFM) phase and gives rise to multiple coexisting magnetic ground states at temperatures close to the bulk Néel temperature (T N ). Furthermore, the AFM phase can be reliably suppressed using modest fields (≈16 mT) from the MFM probe, behaving as a nanoscale magnetic switch. This prototypical study of few-layer CrSBr demonstrates the critical role of layer parity on field-tunable 2D magnetism and validates MFM for use in nanomagnetometry of 2D materials (despite the ubiquitous absence of bulk zero-field magnetism in magnetized sheets).The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adma.202201000.

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Section: Introductionmentioning

confidence: 99%

Visualizing Atomically Layered Magnetism in CrSBr

et al. 2022

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“…The integrals I (l) s , which are Mellin transforms, are generally divergent, and they have to be understood as analytic extensions of their convergent versions. For the Mellin transform in equation (2.6), we note that f (1) (x) + f (2) (x) = 2J ν (bx). Then, from [6, lemma 4, p. 203], we have 3) and performing simplifications, we obtain our main result given by equation (1.3), where the exactifying term is given by We can further simplify equation (1.3) by expanding the trigonometric factors in the first term of equation (3.7).…”

Section: Asymptotic Expansion Of the Hankel Integralmentioning

confidence: 99%

“…The Hankel integral, ∞ 0 f (x)J ν (bx) dx, arises naturally in many fields of application in physics [1][2][3][4][5] example, which has served as the motivation behind this work, arises in the quantum tunnelling problem where the traversal time across a potential barrier appears in the form of a Hankel integral of the zeroth order [5]. On many occasions, it is desirable to obtain an asymptotic estimate of the integral for arbitrarily large b [1,5].…”

Section: Introductionmentioning

confidence: 99%

Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales

Galapon

Martinez

2014

Proc. R. Soc. A.

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We obtain an exactification of the Poincaré asymptotic expansion (PAE) of the Hankel integral,We find that, for halfinteger orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the asymptotic series does not terminate and is generally divergent, but is amenable to superasymptotic summation, i.e. by optimal truncation. For specific examples, we compare the accuracy of the optimally truncated asymptotic series owing to the McClure-Wong distributional method with owing to the Mellin-Barnes integral method. We find that the former is spectacularly more accurate than the latter, by, in some cases, more than 70 orders of magnitude for the same moderate value of b. Moreover, the exactification can lead to a resummation of the PAE when it is exact, with the resummed Poincaré series exhibiting again the same spectacular accuracy. More importantly, the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences.

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“…The various electromagnetic problems involving type-II superconductors, including magnetic levitation force, vortex lattice, etc, have been widely studied when the superconductors have planar boundaries [5][6][7][9][10][11][12][13][14][15][16][17]. Those involving cylindrical superconductors were also studied by many authors [17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Analytic solutions to Maxwell–London equations and levitation force for a general magnetic source in the presence of a type-II superconducting sphere with a vortex line

Lin¹,

Zhao²

2009

Supercond. Sci. Technol.

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The interaction between a general magnetic source and a type-II superconducting sphere with a vortex line is studied within the London theory. With a single vortex line along a diameter of the sphere and a general magnetic source outside the sphere, the Maxwell-London equations are solved analytically. In the source free case, the magnetic field and its flux, the supercurrent and the magnetic moment associated with it, the London free energy of the system, etc are studied in detail. Various cases are investigated and the levitation forces are calculated when the source is a magnetic monopole, a point dipole, a current carrying loop, and a uniform magnetic field. The levitation force contains two parts, one from the stray field of the vortex line and the other from the field of the supercurrent induced by the source. In particular, for a point dipole with arbitrary location and orientation, both parts of the levitation force contain transverse as well as radial components. For a radial or transverse dipole, the levitation force due to the induced field has only a radial component and the results reduce to those previously available. When the dipole is located right above or below the vortex line, the levitation force due to the latter exhibits rather interesting features. At these positions, the two parts of the levitation force on a transverse dipole can be distinguished. Because the model for the vortex line involves a string-like core, the London free energy obtained as a series appears to be divergent in general. An improved model is suggested, where the vortex line has a conical core. Modifications to all results are obtained and the divergence problem is resolved.

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Theory of vortex force microscopy in superconducting layers

Cited by 4 publications

References 30 publications

Visualizing Atomically Layered Magnetism in CrSBr

Visualizing Atomically Layered Magnetism in CrSBr

Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales

Analytic solutions to Maxwell–London equations and levitation force for a general magnetic source in the presence of a type-II superconducting sphere with a vortex line

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